Question

Let Z be a non real complex no. lying on the circle |Z| = 1, then Z is :

• A. tan$\left( \frac{\arg Z}{2} \right)$
• B. sin$\left( \frac{\arg Z}{2} \right)$
• C. $\frac{1 + i\tan\left( \frac{\arg Z}{2} \right)}{1 - i\tan\left( \frac{\arg Z}{2} \right)}$
• D. None of these

Answer 1

Answer: (C)

$\frac{1 + i\tan\left( \frac{\arg Z}{2} \right)}{1 - i\tan\left( \frac{\arg Z}{2} \right)}$

Answer 2

Sol. Since |Z| = 1

let Z = cosa + $i$sina

(where a is argument of Z)

Ž Z = $\frac{1 - \tan^{2}(\alpha/2)}{1 + \tan^{2}(\alpha/2)}$+ $i$ $\frac{2\tan(\alpha/2)}{1 + \tan^{2}(\alpha/2)}$

Ž Z = $\frac{1 - \tan^{2}(\alpha/2) + 2i\tan(\alpha/2)}{1 + \tan^{2}(\alpha/2)}$

Ž Z = $\frac{\left( 1 + i\tan\alpha/2 \right)^{2}}{(1 - i\tan\alpha/2)(1 + i\tan\alpha/2)}$

Ž Z =

Ž $\left( \frac{1 + i\tan(\alpha/2)}{1 - i\tan(\alpha/2)} \right)$

Ž Z = $\frac{1 + itan\left( \frac{Arg.\ Z}{2} \right)}{1–itan\left( \frac{Arg.\ Z}{2} \right)\ }$